LpCase to Systems of Elasticity

In this chapter, the authors study the extended systems of one-dimensional nonlinear elasticity in Lagrangian coordinates with bounded measurable initial data. It is interesting that for the system of elasticity, at almost the same time, there are two different proofs obtained by Lin and J. W. Shearer independently. The authors introduce these proofs as well as an application of Shearer’s proof on an extended system of elasticity of three equations. They also introduce a proof of L^p weak solution existence to the Cauchy problem, which was obtained by Lin by the compensated compactness method coupled with the artificial viscosity approximation. Independent of Lin’s proof, J. W. Shearer considered the L^p solution for the same system of elasticity also with the artificial viscosity approximation, but using different entropy-entropy flux pairs, which were extended late by Serre and Shearer to study the compactness of the physical viscosity approximation.